The priesthood offices of elder, priest, and teacher were different among the Nephites than they are today. First of all, the Nephites did not have the Aaronic Priesthood among them because none of the Nephites or Mulekites were of Levitical descent. Even after Christ appeared to them, there is no scriptural evidence that he established the preparatory priesthood among them (see Footnote). Therefore, all three offices, Elder, Priest, and Teacher, were offices within the Melchizedek priesthood.
In DC 20, the Lord describes the various offices in the priesthood and their individual functions. However, the Book of Mormon does not contain a passage which explicitly describes the different offices and their functions. Therefore, we are left to piece together the structure of the priesthood hierarchy. The four offices within the Melchizedek Priesthood, as practiced among the Nephites, are discussed in hierarchical order:
High Priest: This calling is spoken of from the days of Alma to the coming of Christ. It is the same office held by the ancient Patriarchs (see Alma 13:1-14, DC 107:10). First mentioned in connection with Alma, the elder (Mosiah 23:16), the office was held by the supreme leader of the church. His calling was analogous to that of the President of the Church. While initially, there was only one High Priest, eventually, others were given the same office in conjunction with their ecclesiastical jurisdictions (see Alma 30:20; 46:6). Finally, the office had become corrupted by wickedness (3 Nephi 6:20-21) and was not mentioned after Christ's visit.
Elder: Elders were subordinate to the high priest, as they were ordained by him (Alma 4:7; 6:1). Although mentioned infrequently prior to Christ's visit, they were responsible for presiding and watching over the church (Alma 6:1), ordaining Priests and Teachers (Moro 3:1), administering the sacrament (Moro 4:1), and overseeing church discipline (Moro 6:7).
Priest: Priests were ordained by the Elders or the High Priest (Mosiah 18:18, Moro 3:1). They are most frequently mentioned in connection with Teachers. The Book of Mormon seems to use the terms ordain, consecrate, and appoint in a synonymous fashion when speaking of the calling of Priests and Teachers (2 Nephi 5:26, Jarom 1:11, Mosiah 23:17; 25:19, Alma 4:7; 15:13; 23:4; 45:23, Moro 3:1). The frequent mention of this office signifies that both Priests and Teachers were commonly ordained and were the "workhorses" of the priesthood. Alma ordained a Priest for every 50 members of the church (Mosiah 18:18). Their duties were to preside and watch over the church, preach repentance, teach the gospel, assist in matters of church discipline, and administer the sacrament (Alma 6:1, Moroni 3:3, Mosiah 18:18; 26:7, Moro 4:1). Their calling was greater than that of a teacher because teachers brought sinners to the priests for judgment (Mosiah 26:7).
Teacher: Teachers were ordained by the Elders or the High Priest (Alma 15:13, Moro 3:1). They are most frequently mentioned in connection with Priests (2 Nephi 5:26, Jarom 1:11, Mosiah 23:17; 25:19, Alma 4:7; 15:13; 23:4; 45:23, Moro 3:1). The frequent mention of this office signifies that both Priests and Teachers were commonly ordained and were the "workhorses" of the priesthood. Their duties, other than to teach and preach, are not mentioned. They were subordinate to teachers (Mosiah 26:7), and, interestingly, were not mentioned as officiators in the administration of the sacrament (Moro 4:1).
"The priests and teachers referred to throughout the Book of Mormon were often two distinct groups, even though the book often attributes teaching functions to its priests. The terms priests and teachers are mentioned in close proximity of one another twenty-two times in the Book of Mormon, and in every instance except one 'teachers' are mentioned after 'priests,' suggesting that they might represent a subordinate priesthood office among the Nephites as they do in the Church today." (Daniel C. Peterson, Book of Mormon Symposium Series, Mosiah, edited by PR Cheesman, MS Nyman, and CD Tate, Jr., 1988, p. 195)
Bruce R. McConkie
"Even during the Mosaic dispensation there were no Aaronic Priesthood holders among the Nephites, for there were no Levites among them, and the Aaronic Priesthood in that day was confined to the sons of Levi. The priests and teachers among the Nephites held the Melchizedek Priesthood." (A New Witness for the Articles of Faith, p. 348)